Determine all possible pair(s) of nonnegative integers (P, Q) that satisfy this equation.

                             P⁴ + (P+1)⁴ = Q² + (Q+1)²

Prove that these are the only pair(s) that exist.

Expand each side of the equation:
2P^4 + 4P^3 + 6P^2 + 4P + 1 = 2Q^2 + 2Q + 1

Multiply each side by 2 then subtract 1 from each side:
4P^4 + 8P^3 + 12P^2 + 8P + 1 = 4Q^2 + 4Q + 1

Express each side as an expression squared:
(2P^2 + 2P + 2)^2 - 3 = (2Q+1)^2

The only perfect squares to fill the equation X^2-3=Y^2 are X=2 and Y=1.

Then 2P^2 + 2P + 2 = 2 and 2Q + 1 = 1, which makes P=0 and Q=0.

Posted by Brian Smith on 2010-01-09 17:54:44